Simplify (5+4i)(5-4i)

$(5 + 4 i) (5 - 4 i)$

Step 1

Expand $(5 + 4 i) (5 - 4 i)$ using the FOIL Method.

Apply the distributive property.

$5 (5 - 4 i) + 4 i (5 - 4 i)$

Apply the distributive property.

$5 \cdot 5 + 5 (- 4 i) + 4 i (5 - 4 i)$

Apply the distributive property.

$5 \cdot 5 + 5 (- 4 i) + 4 i \cdot 5 + 4 i (- 4 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 (- 4 i) + 4 i \cdot 5 + 4 i (- 4 i)$

Multiply $- 4$ by $5$ .

$25 - 20 i + 4 i \cdot 5 + 4 i (- 4 i)$

Multiply $5$ by $4$ .

$25 - 20 i + 20 i + 4 i (- 4 i)$

Multiply $4 i (- 4 i)$ .

Multiply $- 4$ by $4$ .

$25 - 20 i + 20 i - 16 i i$

Raise $i$ to the power of $1$ .

$25 - 20 i + 20 i - 16 (i^{1} i)$

Raise $i$ to the power of $1$ .

$25 - 20 i + 20 i - 16 (i^{1} i^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$25 - 20 i + 20 i - 16 i^{1 + 1}$

Add $1$ and $1$ .

$25 - 20 i + 20 i - 16 i^{2}$

Rewrite $i^{2}$ as $- 1$ .

$25 - 20 i + 20 i - 16 \cdot - 1$

Multiply $- 16$ by $- 1$ .

$25 - 20 i + 20 i + 16$

Add $25$ and $16$ .

$41 - 20 i + 20 i$

Add $- 20 i$ and $20 i$ .

$41 + 0$

Add $41$ and $0$ .

$41$